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Pontryagin duality
1 Pontryagin dual
Let be a locally compact abelian group and the 1-torus, i.e. the unit circle in .
Definition - A continuous homomorphism is called a character of . The set of all characters is called the Pontryagin dual of and is denoted by .
Under pointwise multiplication is also an abelian group. Since is a group of functions we can make it a topological group under the compact-open topology (topology of convergence on compact sets).
2 Examples
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, via with .
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, via with .
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, via with .
3 Properties
The following are some important properties of the dual group:
Theorem - Let be a locally compact abelian group. We have that
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is also locally compact.
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is second countable if and only if is second countable.
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is compact if and only if is discrete.
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is discrete if and only if is compact.
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for any finite set . This isomorphism is natural.
4 Pontryagin duality
Let be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map defined by
This canonical construction preserves identity mappings and compositions, i.e. the dualization process is a functor:
Theorem - The dualization is a contravariant functor from the category of locally compact abelian groups to itself.
5 Isomorphism with the second dual
Although in general there is not a canonical identification of with its dual , there is a natural isomorphism between and its dualβs dual :
Theorem - The map defined by , where , is a natural isomorphism between and .
6 Applications
The study of dual groups allows one to visualize Fourier series, Fourier transforms and discrete Fourier transforms from a more abstract and unified view-point, providing the basis for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the foundations of the theory of abstract abelian harmonic analysis.
Mathematics Subject Classification
43A40 Character groups and dual objects22B05 General properties and structure of LCA groups
22D35 Duality theorems
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